Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.8% → 98.6%

Time: 7.8s

Alternatives: 10

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+239}:\\ \;\;\;\;\frac{{\left(x\_m \cdot \mathsf{fma}\left(z, z, 1\right)\right)}^{-1}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot z\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e+239)
     (/ (pow (* x_m (fma z z 1.0)) -1.0) y_m)
     (/ 1.0 (* (* x_m z) (* y_m z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+239) {
		tmp = pow((x_m * fma(z, z, 1.0)), -1.0) / y_m;
	} else {
		tmp = 1.0 / ((x_m * z) * (y_m * z));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+239)
		tmp = Float64((Float64(x_m * fma(z, z, 1.0)) ^ -1.0) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * z) * Float64(y_m * z)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+239], N[(N[Power[N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * z), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000007e239

   1. Initial program 97.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right)}^{-1}}{y}} \]

   if 5.00000000000000007e239 < (*.f64 z z)

   1. Initial program 75.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

      7. lower-*.f64 75.7

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

   5. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{1}{\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+239}:\\ \;\;\;\;\frac{{\left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}^{-1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 + z \cdot z\right) \cdot y\_m \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(y\_m \cdot z, z, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot z\right) \cdot y\_m\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (+ 1.0 (* z z)) y_m) 5e+307)
     (/ (/ 1.0 x_m) (fma (* y_m z) z y_m))
     (/ 1.0 (* (* (* x_m z) y_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((1.0 + (z * z)) * y_m) <= 5e+307) {
		tmp = (1.0 / x_m) / fma((y_m * z), z, y_m);
	} else {
		tmp = 1.0 / (((x_m * z) * y_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(z * z)) * y_m) <= 5e+307)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z) * y_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], 5e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * z), $MachinePrecision] * y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e307

   1. Initial program 95.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot \left(1 + z \cdot z\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(1 + z \cdot z\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z + 1\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot \left(z \cdot z\right) + y \cdot 1}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z\right)} + y \cdot 1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(y \cdot z\right) \cdot z} + y \cdot 1} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\frac{1}{x}}{\left(y \cdot z\right) \cdot z + \color{blue}{y}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(y \cdot z, z, y\right)}} \]

      9. lower-*.f64 97.9

         \[\leadsto \frac{\frac{1}{x}}{\mathsf{fma}\left(\color{blue}{y \cdot z}, z, y\right)} \]

   4. Applied rewrites 97.9%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(y \cdot z, z, y\right)}} \]

   if 5e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 66.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

      7. lower-*.f64 66.6

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.9%

      \[\leadsto \frac{1}{\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\left(\left(x \cdot z\right) \cdot y\right) \cdot \color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z \cdot z\right) \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot z\right) \cdot y\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+239}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot z\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 5e+239)
     (/ (/ (/ 1.0 x_m) (fma z z 1.0)) y_m)
     (/ 1.0 (* (* x_m z) (* y_m z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+239) {
		tmp = ((1.0 / x_m) / fma(z, z, 1.0)) / y_m;
	} else {
		tmp = 1.0 / ((x_m * z) * (y_m * z));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+239)
		tmp = Float64(Float64(Float64(1.0 / x_m) / fma(z, z, 1.0)) / y_m);
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * z) * Float64(y_m * z)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+239], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * z), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000007e239

   1. Initial program 97.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{-y}} \]

   if 5.00000000000000007e239 < (*.f64 z z)

   1. Initial program 75.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

      7. lower-*.f64 75.7

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

   5. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{1}{\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+239}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 + z \cdot z\right) \cdot y\_m \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y\_m, z \cdot z, y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot z\right) \cdot y\_m\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (+ 1.0 (* z z)) y_m) 5e+307)
     (/ 1.0 (* (fma y_m (* z z) y_m) x_m))
     (/ 1.0 (* (* (* x_m z) y_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((1.0 + (z * z)) * y_m) <= 5e+307) {
		tmp = 1.0 / (fma(y_m, (z * z), y_m) * x_m);
	} else {
		tmp = 1.0 / (((x_m * z) * y_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(z * z)) * y_m) <= 5e+307)
		tmp = Float64(1.0 / Float64(fma(y_m, Float64(z * z), y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z) * y_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], 5e+307], N[(1.0 / N[(N[(y$95$m * N[(z * z), $MachinePrecision] + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * z), $MachinePrecision] * y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 5e307

   1. Initial program 95.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{y \cdot \left(1 + z \cdot z\right)} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}} \]

      6. lower-*.f64 95.1

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \left(1 + z \cdot z\right)\right)} \cdot x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{1}{\left(y \cdot \color{blue}{\left(1 + z \cdot z\right)}\right) \cdot x} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right) \cdot x} \]

      10. distribute-lft-in N/A

          \[\leadsto \frac{1}{\color{blue}{\left(y \cdot \left(z \cdot z\right) + y \cdot 1\right)} \cdot x} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{1}{\left(y \cdot \left(z \cdot z\right) + \color{blue}{y}\right) \cdot x} \]

      12. lower-fma.f64 95.1

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(y, z \cdot z, y\right)} \cdot x} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(y, z \cdot z, y\right) \cdot x}} \]

   if 5e307 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 66.6%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

      7. lower-*.f64 66.6

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.9%

      \[\leadsto \frac{1}{\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\left(\left(x \cdot z\right) \cdot y\right) \cdot \color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z \cdot z\right) \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(y, z \cdot z, y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot z\right) \cdot y\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{\left(-y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot z\right) \cdot z\right) \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 0.1)
     (/ (fma z z -1.0) (* (- y_m) x_m))
     (/ 1.0 (* (* (* x_m z) z) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.1) {
		tmp = fma(z, z, -1.0) / (-y_m * x_m);
	} else {
		tmp = 1.0 / (((x_m * z) * z) * y_m);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.1)
		tmp = Float64(fma(z, z, -1.0) / Float64(Float64(-y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z) * z) * y_m));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.1], N[(N[(z * z + -1.0), $MachinePrecision] / N[((-y$95$m) * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * z), $MachinePrecision] * z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2}}{x \cdot y}} + \frac{1}{x \cdot y} \]

      2. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2} + 1}{x \cdot y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1 \cdot {z}^{2} + 1}{\color{blue}{y \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot {z}^{2} + 1}{y}}}{x} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left({z}^{2}\right)\right)} + 1}{y}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot z}\right)\right) + 1}{y}}{x} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot z} + 1}{y}}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), z, 1\right)}}{y}}{x} \]

      11. lower-neg.f64 98.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{-z}, z, 1\right)}{y}}{x} \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-z, z, 1\right)}{y}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left(z, z, -1\right)}{\color{blue}{\left(-y\right) \cdot x}} \]

   if 0.10000000000000001 < (*.f64 z z)

   1. Initial program 82.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(z, z, 1\right) \cdot x\right)}^{-1}}{y}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left({z}^{2} \cdot y\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot {z}^{2}\right) \cdot y}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot {z}^{2}\right) \cdot y}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot x\right)} \cdot y} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot x\right)} \cdot y} \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot x\right) \cdot y} \]

      8. lower-*.f64 76.9

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot x\right) \cdot y} \]

   7. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot x\right) \cdot y}} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.0%

      \[\leadsto \frac{1}{\left(\left(x \cdot z\right) \cdot z\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{\left(-y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot z\right) \cdot y\_m\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 0.1)
     (/ (fma z z -1.0) (* (- y_m) x_m))
     (/ 1.0 (* (* (* x_m z) y_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.1) {
		tmp = fma(z, z, -1.0) / (-y_m * x_m);
	} else {
		tmp = 1.0 / (((x_m * z) * y_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.1)
		tmp = Float64(fma(z, z, -1.0) / Float64(Float64(-y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * z) * y_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.1], N[(N[(z * z + -1.0), $MachinePrecision] / N[((-y$95$m) * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * z), $MachinePrecision] * y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2}}{x \cdot y}} + \frac{1}{x \cdot y} \]

      2. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2} + 1}{x \cdot y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1 \cdot {z}^{2} + 1}{\color{blue}{y \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot {z}^{2} + 1}{y}}}{x} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left({z}^{2}\right)\right)} + 1}{y}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot z}\right)\right) + 1}{y}}{x} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot z} + 1}{y}}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), z, 1\right)}}{y}}{x} \]

      11. lower-neg.f64 98.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{-z}, z, 1\right)}{y}}{x} \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-z, z, 1\right)}{y}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left(z, z, -1\right)}{\color{blue}{\left(-y\right) \cdot x}} \]

   if 0.10000000000000001 < (*.f64 z z)

   1. Initial program 82.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

      7. lower-*.f64 81.1

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.0%

      \[\leadsto \frac{1}{\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.1%

      \[\leadsto \frac{1}{\left(\left(x \cdot z\right) \cdot y\right) \cdot \color{blue}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{\left(-y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot z\right) \cdot \left(y\_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 0.1)
     (/ (fma z z -1.0) (* (- y_m) x_m))
     (/ 1.0 (* (* x_m z) (* y_m z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.1) {
		tmp = fma(z, z, -1.0) / (-y_m * x_m);
	} else {
		tmp = 1.0 / ((x_m * z) * (y_m * z));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.1)
		tmp = Float64(fma(z, z, -1.0) / Float64(Float64(-y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * z) * Float64(y_m * z)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.1], N[(N[(z * z + -1.0), $MachinePrecision] / N[((-y$95$m) * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * z), $MachinePrecision] * N[(y$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2}}{x \cdot y}} + \frac{1}{x \cdot y} \]

      2. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2} + 1}{x \cdot y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1 \cdot {z}^{2} + 1}{\color{blue}{y \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot {z}^{2} + 1}{y}}}{x} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left({z}^{2}\right)\right)} + 1}{y}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot z}\right)\right) + 1}{y}}{x} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot z} + 1}{y}}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), z, 1\right)}}{y}}{x} \]

      11. lower-neg.f64 98.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{-z}, z, 1\right)}{y}}{x} \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-z, z, 1\right)}{y}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left(z, z, -1\right)}{\color{blue}{\left(-y\right) \cdot x}} \]

   if 0.10000000000000001 < (*.f64 z z)

   1. Initial program 82.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

      7. lower-*.f64 81.1

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.0%

      \[\leadsto \frac{1}{\left(z \cdot y\right) \cdot \color{blue}{\left(z \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{\left(-y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{\left(-y\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y\_m \cdot z\right) \cdot x\_m\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* z z) 0.1)
     (/ (fma z z -1.0) (* (- y_m) x_m))
     (/ 1.0 (* (* (* y_m z) x_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.1) {
		tmp = fma(z, z, -1.0) / (-y_m * x_m);
	} else {
		tmp = 1.0 / (((y_m * z) * x_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.1)
		tmp = Float64(fma(z, z, -1.0) / Float64(Float64(-y_m) * x_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * z) * x_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.1], N[(N[(z * z + -1.0), $MachinePrecision] / N[((-y$95$m) * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(y$95$m * z), $MachinePrecision] * x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2}}{x \cdot y}} + \frac{1}{x \cdot y} \]

      2. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2} + 1}{x \cdot y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1 \cdot {z}^{2} + 1}{\color{blue}{y \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-1 \cdot {z}^{2} + 1}{y}}}{x} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left({z}^{2}\right)\right)} + 1}{y}}{x} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot z}\right)\right) + 1}{y}}{x} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot z} + 1}{y}}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), z, 1\right)}}{y}}{x} \]

      11. lower-neg.f64 98.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{-z}, z, 1\right)}{y}}{x} \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-z, z, 1\right)}{y}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \frac{\mathsf{fma}\left(z, z, -1\right)}{\color{blue}{\left(-y\right) \cdot x}} \]

   if 0.10000000000000001 < (*.f64 z z)

   1. Initial program 82.3%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot {z}^{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left({z}^{2} \cdot y\right)} \cdot x} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

      7. lower-*.f64 81.1

         \[\leadsto \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot x} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(z \cdot z\right) \cdot y\right) \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.1%

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(z \cdot y\right)\right) \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, z, -1\right)}{\left(-y\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(y \cdot z\right) \cdot x\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \frac{\frac{1}{y\_m}}{x\_m}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (/ 1.0 y_m) x_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((1.0 / y_m) / x_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((1.0d0 / y_m) / x_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((1.0 / y_m) / x_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * ((1.0 / y_m) / x_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(1.0 / y_m) / x_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * ((1.0 / y_m) / x_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.5%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{{z}^{2}}{x \cdot y} + \frac{1}{x \cdot y}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2}}{x \cdot y}} + \frac{1}{x \cdot y} \]

   2. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot {z}^{2} + 1}{x \cdot y}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-1 \cdot {z}^{2} + 1}{\color{blue}{y \cdot x}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {z}^{2} + 1}{y}}{x}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot {z}^{2} + 1}{y}}}{x} \]

   7. mul-1-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left({z}^{2}\right)\right)} + 1}{y}}{x} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot z}\right)\right) + 1}{y}}{x} \]

   9. distribute-lft-neg-in N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot z} + 1}{y}}{x} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z\right), z, 1\right)}}{y}}{x} \]

   11. lower-neg.f64 47.6

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{-z}, z, 1\right)}{y}}{x} \]

5. Applied rewrites 47.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-z, z, 1\right)}{y}}{x}} \]

6. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{1}{y}}{x} \]
7. Step-by-step derivation

8. Applied rewrites 54.2%

   \[\leadsto \frac{\frac{1}{y}}{x} \]
9. Add Preprocessing

Alternative 10: 58.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \frac{1}{y\_m \cdot x\_m}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ 1.0 (* y_m x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (1.0 / (y_m * x_m)));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * (1.0d0 / (y_m * x_m)))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (1.0 / (y_m * x_m)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * (1.0 / (y_m * x_m)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(1.0 / Float64(y_m * x_m))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * (1.0 / (y_m * x_m)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.5%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{x \cdot y}} \]
4. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]

   3. lower-/.f64 54.2

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{y} \]

5. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]
6. Step-by-step derivation

7. Applied rewrites 54.2%

   \[\leadsto \frac{1}{\color{blue}{y \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t\_0\\ t_2 := \frac{\frac{1}{y}}{t\_0 \cdot x}\\ \mathbf{if}\;t\_1 < -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))